A DETERMINACY APPROACH TO BOREL

COMBINATORICS

ANDREW S. MARKS

Abstract. We introduce a new method, involving infinite games andBorel determinacy, which we use to answer several well-known questionsin Borel combinatorics.

1. Introduction

A Borel graph on a standard Borel space X is a symmetric irreflexiverelation G on X that is Borel as a subset of X ×X. We call elements of Xvertices, and if x, y ∈ X and x G y then we say that x and y are neighbors,or are adjacent. The degree of a vertex is its number of neighbors, anda graph is said to have degree ≤ n if each of its vertices has degree ≤ n.A graph is said to be regular if all of its vertices have the same number ofneighbors, and is n-regular if this number is n.

Graph coloring is a typical problem studied in the field of Borel com-binatorics, where a Borel coloring of a Borel graph G on X is a Borelfunction c : X → Y from the vertices of G to a standard Borel space Y suchthat if x G y, then c(x) 6= c(y). The Borel chromatic number χB(G)of G is the least cardinality of a standard Borel space Y such that G has aBorel coloring with codomain Y . The first systematic study of Borel chro-matic numbers was done by Kechris, Solecki, and Todorcevic [16]. Sincethen, fruitful connections have been found between the study of Borel chro-matic numbers and other areas of mathematics such as ergodic theory anddynamics [3, 4], and dichotomies in descriptive set theory [23].

IfG is a Borel graph, then it is clear that χ(G) ≤ χB(G), where χ(G) is theusual chromatic number of G. However, χ(G) and χB(G) may differ quitewildly. For instance, Kechris, Solecki, and Todorcevic [16] show the existenceof an acyclic Borel graph G0 (so χ(G0) = 2) for which χB(G0) = 2ℵ0 .Nevertheless, in some respects the Borel chromatic number of a graph isquite similar to the usual chromatic number. For example, we have thefollowing analogue of an obvious classical fact:

2010 Mathematics Subject Classification. Primary 03E15.The author is partially supported by the National Science Foundation under DMS-

1204907, and the John Templeton foundation under Award No. 15619.The author would also like to thank the Institute for Mathematical Sciences and the

Department of Mathematics of the National University of Singapore and the John Tem-pleton Foundation for their support to attend the 2012 summer school in logic, where themain lemma of this paper was conceived.

1

2 ANDREW S. MARKS

Theorem 1.1 (Kechris, Solecki, and Todorcevic [16]). If G is a Borel graphof degree ≤ n, then χB(G) ≤ n+ 1.

We will be interested in Borel graphs that arise from free Borel actionsof countable marked groups. Recall that a marked group is a group witha specified set of generators. We assume throughout this paper that the setof generators of a group does not include the identity. Let Γ be a countablediscrete group, and X be a standard Borel space. We endow the space XΓ

of functions from Γ to X with the usual product Borel structure (arisingfrom the product topology) so that XΓ is also a standard Borel space. Theleft shift action of Γ on XΓ is defined by α · y(β) = y(α−1β) for y ∈ XΓ

and α, β ∈ Γ. The free part of this action, denoted Free(XΓ), is the setof y ∈ XΓ such that γ · y 6= y for all nonidentity γ ∈ Γ. Now we defineG(Γ, X) to be the Borel graph on Free(XΓ) where for x, y ∈ Free(XΓ), wehave x G(Γ, X) y if there is a generator γ ∈ Γ such that γ ·x = y or γ ·y = x.Hence, each connected component of G(Γ, X) is an isomorphic copy of theCayley graph of Γ. We will only be interested in G(Γ, X) when Γ is finitelygenerated; an easy Baire category argument shows that if Γ has infinitelymany generators, then χB(G(Γ, 2)) = 2ℵ0 (see [16]).

If Γ is a marked countable group, then G(Γ,N) attains the maximumBorel chromatic number of all graphs generated by a free Borel action ofΓ. That is, suppose we have any free Borel action of Γ on a standardBorel space X, to which we associate the Borel graph GXΓ on X wherex GXΓ y if there is a generator γ of Γ such that γ · x = y or γ · y = x.Then χB(GXΓ ) ≤ χB(G(Γ,N)). This is trivial when Γ is finite. When Γis infinite, it follows from [15, Theorem 5.4]; since the action of Γ on Xis free, the function constructed there will an injective equivariant functionfrom X into Free(NΓ). Recall that if Γ acts on the spaces X and Y , then afunction f : X → Y is said to be equivariant if for all γ ∈ Γ we have thatγ · f(x) = f(γ · x).

Our first result is a theorem describing how the Borel chromatic numberof G(Γ,N) behaves with respect to free products (see 3.1). We stipulate thatif Γ and ∆ are marked groups, then their free product Γ ∗∆ is the markedgroup generated by the union of the generators of Γ and ∆.

Theorem 1.2. If Γ and ∆ are finitely generated marked groups, then

χB(G(Γ ∗∆,N)) ≥ χB(G(Γ,N)) + χB(G(∆,N))− 1

It has been an open question what Borel chromatic numbers can be at-tained by an n-regular acyclic Borel graph, and whether the upper boundgiven by Theorem 1.1 is optimal for such graphs. Several prior results existalong these lines. For 2-regular acyclic graphs, we have that χB(G(Z, 2)) =3 by [16]. More recently, Conley and Kechris [3] have shown that for

the free group on n generators, χB(G(Fn, 2)) ≥ n+2√n−1

2√n−1

, and Lyons and

Nazarov [20] have pointed out that results of Frieze and Luczak [12] implythat χB(G(Fn, 2)) ≥ n

log 2n for sufficiently large n.

A DETERMINACY APPROACH TO BOREL COMBINATORICS 3

Using Theorem 1.2, we answer this question and show that for every nthere exists an n-regular acyclic Borel graph with Borel chromatic numberequal to n+ 1. Indeed, if (Z/2Z)∗n is the free product of n copies of Z/2Z,then χB (G ((Z/2Z)∗n,N)) = n + 1, since Theorem 1.2 gives a tight lowerbound to the upper bound of Theorem 1.1. Similarly, for the free group onn generators, we have χB(G(Fn,N)) = 2n+ 1.

Further, we can give a complete description of the Borel chromatic num-bers that can be attained by an n-regular acyclic Borel graph; they areexactly those allowed by Theorem 1.1 (see 3.3):

Theorem 1.3. For every n ≥ 1 and every m ∈ {2, . . . , n + 1}, there is an-regular acyclic Borel graph G with χB(G) = m.

In the theorem above, G may be chosen to arise from a free Borel actionof (Z/2Z)∗n.

Our results above involve graphs of the form G(Γ,N). Answering a ques-tion originally posed in an early version of this paper, Seward and Tucker-Drob [24] have shown that for all marked groups Γ, and all n ≥ 2, we haveχB(G(Γ,N)) = χB(G(Γ, n)). Hence, our results apply to graphs of the formG(Γ, 2) as well.

Next, we turn to Borel edge colorings. Let G be a Borel graph on astandard Borel space X. If x, y ∈ X then we say the set {x, y} is an edge ofG if x G y. The line graph G of G is the graph whose vertices are the edgesof G, and where distinct {x, y} and {z, w} are adjacent if {x, y}∩{z, w} 6= ∅.A Borel edge coloring of G is defined to be a Borel coloring of G. TheBorel edge chromatic number of a Borel graph G, denoted χ′B(G), isthe Borel chromatic number of its line graph.

It is a classical theorem of Vizing (see e.g. [9, Theorem 5.3.2]) that everyn-regular graph has an edge coloring with n + 1 colors. Kechris, Soleckiand Todorcevic have asked if the analogous fact is true for n-regular Borelgraphs [16, page 15]. More recently, this question has attracted some interestfrom the study of graph limits [11] [14, Remark 3.8]. We show that thisquestion has a negative answer, and we calculate exactly what Borel edgechromatic numbers can be attained by an n-regular Borel graph. Note thatif G is an n-regular Borel graph, then since G is 2n− 2 regular, we see thatχB(G) ≤ 2n − 1 by Theorem 1.1. We show that this obvious upper boundcan be achieved, even using acyclic and Borel bipartite graphs (see 3.10).Recall that a Borel bipartite graph is a Borel graph G on X for whichthere is a partition of X into two Borel sets A and B such that if x G y,then either x ∈ A and y ∈ B, or x ∈ B and y ∈ A.

Theorem 1.4. For every n ≥ 1 and every m ∈ {n, . . . , 2n− 1}, there is ann-regular acyclic Borel bipartite graph G such that χ′B(G) = m.

A Borel perfect matching of a Borel graph G is a Borel subset M ofthe edges of G such that every vertex of G is incident to exactly one edge ofM . In his 1993 problem list, Miller asked whether there is a Borel analogue

4 ANDREW S. MARKS

of Hall’s theorem for matchings [22, 15.10]. Laczkovich [19] showed theexistence of a 2-regular Borel bipartite graph with no Borel perfect matching,and this result was extended to give examples of n-regular Borel bipartitegraphs with no Borel perfect matchings by Conley and Kechris [3] when nis even. However, the case for odd n > 1 had remained open. We obtainthe following (see 3.8):

Theorem 1.5. For every n > 1, there exists an n-regular acyclic Borelbipartite graph with no Borel perfect matching.

Some positive results on measurable matchings have recently been ob-tained by Lyons and Nazarov [20]. Among their results, they show that thegraph we use to prove the case n = 3 in Theorem 1.5 has a Borel matchingmodulo a null set with respect to a natural measure. Further work on match-ings in the measurable context has been done by Csoka and Lippner [8]. Themeasurable analogue of Theorem 1.5 for odd n remains open.

Both Theorems 1.4 and 1.5 are corollaries of the following result on Boreldisjoint complete sections (see 3.7). Suppose X is a standard Borel space,and E is an equivalence relation on X. Then a complete section for E isa set A ⊆ X that meets every equivalence class of E. Now suppose that Fis also an equivalence relation on X. Then say that E and F have Boreldisjoint complete sections if there exist disjoint Borel sets A,B ⊆ Xsuch that A is a complete section for E and B is a complete section for F .

Theorem 1.6. Let Γ and ∆ be countable groups. Let EΓ be the equivalencerelation on Free(NΓ∗∆) where x EΓ y if there exists a γ ∈ Γ such thatγ · x = y. Define E∆ analogously. Then EΓ and E∆ do not have Boreldisjoint complete sections.

Theorems 1.2-1.6 above all follow from a single lemma which we prove inSection 2. Unusually for the subject, this lemma is proved using a directapplication of Borel determinacy. Borel determinacy is the theorem, due toMartin [21], that there is a winning strategy for one of the players in everyinfinite two-player game of perfect information with a Borel payoff set. Wewill use the determinacy of a class of games for constructing functions fromfree products of countable groups to N. Thus, we are also interested indifferences between the results proved using our new technique, and whatcan be shown using more standard tools such as measure theory and Bairecategory, which have been a mainstay of proofs in Borel combinatorics.

Here, Theorem 1.6 provides a nice contrast because it is not true in thecontext of measure or category, except for the single case where Γ = ∆ =Z/2Z. Indeed, we have the following more general theorem (see 4.5). Recallthat a countable Borel equivalence relation on a standard Borel spaceX is an equivalence relation on X that is Borel as a subset of X ×X andwhose equivalence classes are countable. EΓ and E∆ in Theorem 1.6 areexamples of countable Borel equivalence relations.

A DETERMINACY APPROACH TO BOREL COMBINATORICS 5

Theorem 1.7. Suppose E and F are countable Borel equivalence relationson a standard Borel space X such that every equivalence class of E hascardinality ≥ 3 and every equivalence class of F has cardinality ≥ 2. ThenE and F have Borel disjoint complete sections modulo a null set or meagerset with respect to any Borel probability measure on X or Polish topologyrealizing the standard Borel structure of X.

As we will see, the idea of disjoint complete sections turns out to be sur-prisingly robust, as evidenced by a large number of equivalent formulationswhich we give in Theorems 4.5 and 4.7 later in the paper. Using the exis-tence of disjoint complete sections in the context of measure and category,we also show the following, which contrasts nicely with Theorem 1.4, anddemonstrates that it can not be proved using measure-theoretic or Bairecategory techniques (see 4.8):

Theorem 1.8. Suppose G is a 3-regular Borel bipartite graph on X. ThenG has a Borel edge coloring with 4 colors modulo a null set or meager set withrespect to any Borel probability measure on X or Polish topology realizingthe standard Borel structure of X.

Finally, in recent joint work with Clinton Conley and Robin Tucker-Drob[7], we have shown that for every n ≥ 3 and every Borel graph Gof degree ≤ n on a standard Borel space X, if G does not contain a com-plete graph on n+ 1 vertices, then there is a µ-measurable n-coloring of Gwith respect to any Borel probability measure µ on X and a Baire measur-able n-coloring of G with respect to every compatible Polish topology onX. Hence, Theorem 1.3 can not be proved using pure measure theoretic orBaire category arguments, except in the exceptional case n = 2.

1.1. Notation and conventions. Our basic reference for descriptive settheory is [17]. Throughout we will use X, Y , and Z to denote standardBorel spaces, x, y, and z for elements of such spaces, and A, B, and C forsubsets of standard Borel spaces (which will generally be Borel). Given asubset A of a standard Borel space, we let Ac denote its complement.

We will use E and F for countable Borel equivalence relations, and G andH for Borel graphs. We will use f , g, and h to denote functions betweenstandard Borel spaces, and c for Borel colorings. Γ and ∆ will be used todenote countable groups, and α, β, γ, and δ will be their elements. We willuse e for the identity of a group. By countable group, we will always meancountable discrete group.

If E is a countable Borel equivalence relation on a standard Borel spaceX, then A ⊆ X is said to be E-invariant if x ∈ A and x E y implies y ∈ A.If B is a subset of X, then we will often consider the largest E-invariantsubset of B. Precisely, this is the set A of x ∈ X such that for all y ∈ Xwhere y E x, we have y ∈ B.

1.2. Acknowledgments. Section 4 of this paper is taken from the author’sthesis, which was written under the excellent direction of Ted Slaman. The

6 ANDREW S. MARKS

author would like to thank Professor Slaman for many years of wise advice.The author would also like to thank Clinton Conley, Alekos Kechris, Ben-jamin Miller, Anush Tserunyan, Robin Tucker-Drob, and Jay Williams forproviding helpful feedback and suggestions throughout the development ofthis paper. Finally, the author would like to thank the referee for manyhelpful suggestions.

2. The main lemma

Let Γ and ∆ be disjoint countable groups, and let Γ ∗ ∆ be their freeproduct. Each nonidentity element of Γ ∗ ∆ can be uniquely written asa finite product of either the form γi0δi1γi2δi3 . . . or δi0γi1δi2γi3 . . ., whereγi ∈ Γ and δi ∈ ∆ are nonidentity elements for all i. Words of the formerform we call Γ-words, and words of the latter form we call ∆-words. Ourproof will use games for building an element y ∈ NΓ∗∆ where player I definesy on Γ-words and player II defines y on ∆-words.

The following simple observation will let us combine winning strategiesin these games in a useful way. Let WΓ and W∆ be the sets of Γ-words and∆-words respectively. Then for distinct γ, γ′ ∈ Γ we have that γW∆ andγ′W∆ are disjoint, and the analogous fact is true when the roles of Γ and ∆are switched.

We now proceed to our main lemma. Note that both Γ and ∆ act onFree(NΓ∗∆) by restricting the left shift action of Γ ∗∆ to these subgroups.

Lemma 2.1. [Main Lemma] Let Γ,∆ be countable groups. If A ⊆ Free(NΓ∗∆)is any Borel set, then at least one of the following holds:

(1) There is a continuous injective function f : Free(NΓ)→ Free(NΓ∗∆)that is equivariant with respect to the left shift action of Γ on thesespaces and such that ran(f) ⊆ A.

(2) There is an continuous injective function f : Free(N∆)→ Free(NΓ∗∆)that is equivariant with respect to the left shift action of ∆ on thesespaces and such that ran(f) ⊆ Free(NΓ∗∆) \A.

Proof. The main difficulty in our proof is arranging that our games produceelements of Free(NΓ∗∆), and not merely elements of NΓ∗∆. To begin, wemake a definition that will get us halfway there. Let Y be the largestinvariant set of y ∈ NΓ∗∆ such that for all nonidentity γ ∈ Γ and δ ∈ ∆, wehave γ · y 6= y, and δ · y 6= y. That is, Y is the set of x ∈ NΓ∗∆ such that forall α ∈ Γ ∗∆, if y = α−1 · x, then y has the property above. Note that Ycontains Free(NΓ∗∆).

Next, we give a definition that we will use to organize the turn on whichy(α) is defined in our game for each α ∈ Γ∗∆. Fix injective listings γ0, γ1, . . .and δ0, δ1, . . . of all the nonidentity elements of Γ and ∆ respectively. Wedefine the turn function t : Γ ∗∆ → N as follows. First, define t(e) = −1.Then, for each nonidentity element α ∈ Γ ∗ ∆, there is a unique sequencei0, i1 . . . im such that α = γi0δi1γi2 . . . or α = δi0γi1δi2 . . .. We define t(α)

A DETERMINACY APPROACH TO BOREL COMBINATORICS 7

to be the least n such that the associated sequence i0, i1, . . . im for α hasij + j ≤ n for all j ≤ m. The key property of this definition is that if i ≤ n,and α is a ∆-word or the identity, then t(γiα) ≤ n if and only if t(α) < n.Of course, this remains true when the roles of Γ and ∆ are switched.

Now given a Borel set B ⊆ Y , and k ∈ N we define the following game GBkfor producing a y ∈ NΓ∗∆ such that y(e) = k. Player I goes first, and theplayers alternate defining y on finitely many nonidentity elements of Γ ∗∆as follows. On the nth turn of the game for n ≥ 0, player I must define y(α)on all Γ-words α with t(α) = n, and then player II must respond by definingy(α) on all ∆-words α with t(α) = n. We give an illustration of how thegame is played:

I IIy(γ0)

y(δ0)y(γ1)y(γ0δ0)y(γ1δ0)

...

All that remains is to define the winning condition of the game. First, ifthe y that is produced is in Y , then Player II wins the game if and only if yis in B. If y /∈ Y then there must be some α such that there is a nonidentityγ ∈ Γ such that γα−1 · y = α−1 · y, or there is a nonidentity δ ∈ ∆ such thatδα−1 · y = α−1 · y. In the former case, say (α,Γ) witnesses y /∈ Y , and in thelatter say (α,∆) witnesses y /∈ Y . Say α witnesses y /∈ Y if either (α,Γ) or(α,∆) witnesses y /∈ Y . Now if (e,Γ) witnesses y /∈ Y , then player I loses.Otherwise, if (e,∆) witnesses y /∈ Y then player II loses. Finally, if neitherof the above happens, then player I wins if and only if there is a ∆-word αwitnessing y /∈ Y such that for all Γ-words β with t(β) ≤ t(α), we have thatβ does not witness y /∈ Y . This finishes the definition of our game.

Next, we associate to our set A ⊆ Free(NΓ∗∆) a set BA that we will use inthe play of our game. Let EΓ be the equivalence relation on Y where x EΓ yif there is a γ ∈ Γ such that γ · x = y. Define E∆ similarly. By Lemma 2.3which we defer till later, we can find a Borel subset C of Y \ Free(NΓ∗∆)such that C meets every E∆-class on Y \ Free(NΓ∗∆) and its complementCc meets every EΓ-class on Y \ Free(NΓ∗∆). Let BA = A ∪ C. Our use ofC here will be important at the end of the proof to ensure that we create afunction into Free(NΓ∗∆) and not merely into Y .

By Borel determinacy, either player I or player II has a winning strategy

in GBAk for each k ∈ N. So by the pigeon-hole principle, either player I

wins GBAk for infinitely many k or player II wins GBA

k for infinitely manyk. Assume the latter case holds, and let S be the set of k such that player

II wins GBAk . An analogous argument will work in the case that player I

wins for infinitely many k. Since there is a continuous injective equivariantfunction from Free(NΓ) to Free(SΓ), it will suffice to define a continuous

8 ANDREW S. MARKS

injection f : Free(SΓ)→ Free(NΓ∗∆) that is equivariant with respect to theleft shift action of Γ on these spaces and such that ran(f) ⊆ A. Fix winning

strategies in each game GBAk for k ∈ S.

We will define f so that for all x ∈ Free(NΓ) and all γ ∈ Γ, we havef(x)(γ) = x(γ), and so that for all x, f(x) will be a winning outcome of

player II’s winning strategy in the game GBA

x(e).

We proceed as follows. Fix an x in Free(NΓ). For each γ ∈ Γ we will play

an instance of the game GBA

x(γ−1)whose outcome will be γ · f(x). We play

these games for all γ ∈ Γ simultaneously. The moves for player II in thesegames will be made by the winning strategies that we have fixed. We willspecify how to move for player I in these games to satisfy our requirementthat f is equivariant and f(x)(γ) = x(γ).

So for each γ ∈ Γ, we are playing an instance of the game GBA

x(γ−1)to

define a y ∈ NΓ∗∆ equal to γ · f(x). To begin, we have γ · f(x)(e) = x(γ−1)by the definition of the game.

Inductively, suppose γ · f(x)(α) is defined for all γ ∈ Γ and all α witht(α) < n. We need to make the nth move for player I in all our games.Suppose β is a Γ-word with t(β) = n so we can write β = γiα wherei ≤ n and t(α) < n. For all γ ∈ Γ, we now define (γ · f(x))(γiα) =(γ−1i · (γ · f(x)))(α) = (γ−1

i γ · f(x))(α), which has already been defined in

the game associated to γ−1i γ by assumption. Hence we can make the nth

move for player I in all our games using this information. To finish the nthturn, the winning strategies for player II in these games respond with theirnth moves, defining γ · f(x)(δiα) for all i ≤ n and all α such that α = e orα is a ∆-word with t(α) < n.

Based on our definition, it is clear that f is injective, continuous, Γ-equivariant, and that f(x) is an outcome of player II’s winning strategy in

GBA

x(e). All that remains is to show ran(f) ⊆ A.

First, we argue that for all x ∈ Free(NΓ), we have f(x) ∈ Y . Nowsince x ∈ Free(NΓ) and f(x)(e) = x(e), we see that (e,Γ) can not witnessf(x) /∈ Y . Further, since f(x) is a winning outcome of a strategy for playerII, (e,∆) can not witness f(x) /∈ Y . Now we can prove inductively thatα does not witness f(x) /∈ Y for all x ∈ Free(NΓ) and all α ∈ Γ ∗ ∆ witht(α) = n. For each n we do the case of Γ-words first, and then the case of∆-words. Suppose α is a Γ-word with t(α) = n, so α = γβ for some γ ∈ Γand β with t(β) < t(α). Since α−1 · f(x) = β−1γ−1 · f(x) = β−1 · f(γ−1 · x)and β does not witness f(γ−1 ·x) /∈ Y by our induction hypothesis, we musthave that α does not witness f(x) /∈ Y . Now suppose α is a ∆-word witht(α) = n. We may assume no Γ-word β with t(β) ≤ n witnesses f(x) /∈ Y .Hence, we see that player II must ensure α does not witness α · f(x) /∈ Yotherwise they lose the game GBA

x(e) used to define f(x).

A DETERMINACY APPROACH TO BOREL COMBINATORICS 9

For all x, since f(x) ∈ Y , we have f(x) ∈ BA, since f(x) is a winning

outcome for player II in some GBA

x(e). Finally, we claim that f(x) ∈ A for all

x. This is because ran(f) and A are Γ-invariant, BA = A ∪ C, and C doesnot contain any nonempty Γ-invariant sets by definition. �

To finish establishing Lemma 2.1, we must prove Lemma 2.3 which wasused to define the set C above. We will prove a version for countably manyequivalence relations instead of merely two, since we will use this moregeneral version in a later paper.

We begin by recalling a useful tool for organizing constructions in Borelcombinatorics. Let X be a standard Borel space. We let [X]<∞ denotethe standard Borel space of finite subsets of X. If E is a countable Borelequivalence relation, we let [E]<∞ be the Borel subset of [X]<∞ consistingof the S ∈ [X]<∞ such that S is a subset of some equivalence class of E.If Y is a Borel subset of [X]<∞, then the intersection graph on Y is thegraph G where R G S for distinct R,S ∈ Y if R ∩ S 6= ∅.

Lemma 2.2 ([18, Lemma 7.3] [6, Proposition 2]). Suppose E is a countableBorel equivalence relation and let G be the intersection graph on [E]<∞.Then G has a Borel N-coloring.

We will often use this lemma in the following way. Suppose E is a count-able Borel equivalence relation and A is a Borel subset of [E]<∞ containingat least one subset of every E-class. Then there is a Borel set B ⊆ A suchthat elements of B are pairwise disjoint, and B meets every E-class. To seethis, pick some Borel N-coloring of the intersection graph of [E]<∞ usingLemma 2.2, and then let B be the set of R ∈ A that are assigned the leastcolor of all elements of A from the same E-class.

We need a couple more definitions. Suppose that I ∈ {1, 2, . . . ,∞} and{Ei}i<I are finitely many or countably many equivalence relations on X.Then the Ei are said to be non-independent if there exists a sequencex0, x1, . . . , xn of distinct elements of X, and i0, i1, . . . in ∈ N with n ≥ 2such that ij 6= ij+1 for j < n, in 6= i0, and x0 Ei0 x1 Ei1 x2 . . . xn Ein x0.We say this pair of sequences x0, . . . , xn and i0, . . . , in witnesses the non-independence of the Ei. The Ei are said to be independent if they arenot non-independent. The join of the Ei, denoted

∨i<I Ei, is the smallest

equivalence relation containing all the Ei. Precisely, x and y are∨i<I Ei-

related if there is a sequence x0, x1, . . . xn of elements in X such that x = x0,y = xn, and for all j < n, we have xj Ei xj+1 for some i < I. Finally, wesay that the Ei are everywhere non-independent if for every

∨i<I Ei

equivalence class A ⊆ X, the restrictions of the Ei to A are not independent.

Lemma 2.3. Suppose that I ∈ {1, 2, . . . ,∞} and {Ei}i<I are countableBorel equivalence relations on a standard Borel space X that are everywherenon-independent. Then there exists a Borel partition {Ai}i<I of X such thatfor all i < I, Ai

c meets every Ei-class.

10 ANDREW S. MARKS

Proof. Using Lemma 2.2, let C ⊆ [∨i<I Ei]

<∞ be a Borel set containing atleast one subset of each equivalence class of

∨i<I Ei such that the elements of

C are pairwise disjoint and each set {x0, x1, . . . , xn} ∈ C can be assigned anorder x0, x1, . . . , xn and an associated sequence i0, . . . , in of natural numberssuch that these sequences witness the failure of the independence of the Ei.Fix a Borel way of assigning such an order and associated i0, . . . , in to eachelement of C. Define disjoint sets {Ai,0}i<I by setting x ∈ Ai,0 if there is anelement {x0, . . . , xn} of C with associated sequence i0, . . . , in and a j ≤ nsuch that x = xj and i = ij . Note that for all i < I and for all x ∈ Ai,0,there is a y ∈ [x]Ei and j 6= i such that y ∈ Aj,0.

Let k0, k1, . . . be a sequence containing each number less than I infinitelymany times. Given {Ai,n}i<I we construct disjoint sets {Ai,n+1}i<I , whereAi,n+1 ⊇ Ai,n as follows. Let Bi,n+1 be the set of x such that x ∈ [Akn,n]Ei

and for all j < i, x /∈ [Akn,n]Ej . Then let Ai,n+1 = Ai,n∪ (Bi,n+1 \∪j 6=iAj,n).The sets Ai,n are Borel, since every Ei is generated by the Borel action of acountable group by the Feldman-Moore theorem [18, Theorem 1.3].

It is easy to prove by induction that for all x ∈ Ai,n, there is a y ∈ [x]Ei

and a j 6= i such that y ∈ Aj,n. Let Ai = ∪nAi,n, which are disjoint andpartition the space. �

3. Applications to Borel chromatic numbers and matchings

We now show how our main lemma can be applied to prove the theoremsdiscussed in the introduction. Recall that if G and H are Borel graphs on thestandard Borel spaces X and Y respectively, then a Borel homomorphismfrom G to H is a Borel function f : X → Y such that x G y impliesf(x) H f(y). It is clear that if there is a Borel homomorphism f from G toH, then χB(G) ≤ χB(H); if c is a Borel coloring of H, then c ◦ f is a Borelcoloring of G.

Theorem 3.1. If Γ and ∆ are finitely generated marked groups, then

χB(G(Γ ∗∆,N)) ≥ χB(G(Γ,N)) + χB(G(∆,N))− 1

Proof. Suppose χB(G(Γ,N)) = n + 1 and χB(G(∆,N)) = m + 1 so thatG(Γ,N) has no Borel n-coloring and G(∆,N) has no Borel m-coloring. Nowsuppose c : Free(NΓ∗∆)→ {0, 1, . . . , (n+m−1)} was a Borel n+m-coloringof G(Γ ∗ ∆,N) and let A be the set of x such that c(x) < n. If f is thecontinuous equivariant function produced by Lemma 2.1, then c ◦ f giveseither a Borel n-coloring of G(Γ,N) or a Borel m-coloring of G(∆,N), bothof which are contradictions. �

Let C be the class of finitely generated marked groups Γ such that G(Γ,N)is n-regular, and χB(G(Γ,N)) = n + 1, so that the upper bound on theBorel chromatic number of G(Γ,N) given by Theorem 1.1 is sharp. Brooks’stheorem in finite graph theory (see e.g. [9, Theorem 5.2.4]) implies that thefinite groups included in C are exactly those whose Cayley graphs are oddcycles or complete graphs on n vertices. The only prior results giving infinite

A DETERMINACY APPROACH TO BOREL COMBINATORICS 11

groups in C are from [16] where we have that Z and Z/2Z∗Z/2Z are in C whenequipped with their usual generators. Conley and Kechris [3, Theorem 0.10]have shown that these are the only two groups with finitely many ends thatare in C. Theorem 3.1 implies that C is closed under free products; if G(Γ,N)is n-regular and G(∆,N) is m-regular, then G(Γ ∗ ∆,N) is n + m-regular.For example, χB(G((Z/2Z)∗n,N)) = n + 1, and χB(G(Fn,N)) = 2n + 1 forall n.

Next, we will show that the Borel chromatic number of an n-regularacyclic Borel graph can take any of the possible values between 2 and n+ 1allowed by Theorem 1.1.

We begin with an easy lemma.

Lemma 3.2. Suppose G and H are acyclic Borel graphs on the standardBorel spaces X and Y , where χB(G) ≥ 2. Suppose also f : X → Y isan injective Borel homomorphism from G to H such that ∀x, y ∈ X, ifx and y are in different connected components of G, then f(x) and f(y)are in different connected components of H. Then if A = [ran(f)]H is thesaturation of the range of f under the connectedness relation of H, thenχB(G) = χB(H � A).

Proof. χB(G) ≤ χB(H � A) since there is a Borel homomorphism from G toH � A. It remains to show that χB(H � A) ≤ χB(G). Suppose c : X → Zis a Borel coloring of G. Fix two colors z0 and z1 ∈ Z. Now we constructa Borel coloring c′ : A → Z of H � A as follows. If y ∈ ran(f), then letc′(y) = c(f−1(y)). Otherwise, there is a unique path in H of shortest lengthl from y to an element y′ ∈ ran(f). If c(f−1(y′)) = z0, then let c(y) = z1 ifl is odd and c(y) = z0 if l is even. If c(f−1(y′)) 6= z0, then let c(y) = z0 if lis odd and c(y) = z1 if l is even. �

We are ready to proceed.

Theorem 3.3. For every n ≥ 1 and every m ∈ {2, . . . , n + 1}, there is an-regular acyclic Borel graph G with χB(G) = m.

Proof. We have shown that for every k ≥ 2 we have χB(G((Z/2Z)∗k,N)) =

k + 1. Given m ∈ {2, . . . , n + 1}, canonically identify (Z/2Z)∗(m−1) with a

subgroup of (Z/2Z)∗n. Now let f : Free(N(Z/2Z)∗(m−1))→ Free(N(Z/2Z)∗n) be

the function where

f(x)(γ) =

{x(γ) if γ ∈ (Z/2Z)∗(m−1)

0 otherwise.

We finish by applying Lemma 3.2 with G = G((Z/2Z)∗(m−1),N), H =G((Z/2Z)∗n,N) and f as above to obtain a Borel set A saturated under theconnectedness relation of H so H � A is n-regular and χB(H � A) = m. �

The only case we know of where Theorem 3.1 gives a sharp lower boundfor the chromatic number of G(Γ ∗∆,N) is when Γ and ∆ are in the class

12 ANDREW S. MARKS

C we have discussed above. However, it is open whether the lower bound ofTheorem 3.1 can ever be exceeded.

Question 3.4. Are there finitely generated marked groups Γ and ∆ suchthat χB(G(Γ ∗∆,N)) > χB(G(Γ,N)) + χB(G(∆,N))− 1?

Now there is another obvious upper bound on the Borel chromatic numberof G(Γ ∗∆,N) which is better in some cases than that of Theorem 1.1:

Proposition 3.5. If Γ and ∆ are finitely generated marked groups, thenχB(G(Γ ∗∆,N)) ≤ χB(G(Γ,N))χB(G(∆,N))

Proof. We can decompose G(Γ ∗ ∆, X) as the disjoint union of two Borelgraphs GΓ and G∆ given by the edges corresponding to generators of Γ and∆ respectively. Since GΓ and G∆ are induced by free actions of Γ and ∆,their Borel chromatic numbers are less than or equal to χB(G(Γ,N)) andχB(G(∆,N)) and hence we can use pairs of these colors to color G(Γ ∗∆,N).

�

It is likewise open whether this upper bound can ever be achieved.

Question 3.6. Are there nontrivial finitely generated marked groups Γ and∆ such that χB(G(Γ ∗∆,N)) = χB(G(Γ,N))χB(G(∆,N))?

A positive answer to this question would also give a positive answer toQuestion 3.4. It seems natural to believe Question 3.6 has a positive answerin cases where G(Γ ∗∆,N) is n-regular and χB(G(Γ,N))χB(G(∆,N)) ≤ n,so that the bound of Proposition 3.5 is better than that of Theorem 1.1.For example, if m > 2 is even, and we generate Z/mZ by a single element,then G(Z/mZ ∗Z/mZ,N) is 4-regular and has Borel chromatic number ≤ 4by Proposition 3.5. Likewise, Zn is another source of such examples, sinceG(Zn,N) is a 2n-regular Borel graph with χB(G(Zn,N)) ≤ 4 by [13].

Next, we turn to matchings and edge colorings. We begin with the fol-lowing theorem on disjoint complete sections.

Theorem 3.7. Let Γ and ∆ be countable groups. Let EΓ be the equivalencerelation on Free(NΓ∗∆) where x EΓ y if there exists a γ ∈ Γ such thatγ · x = y. Define E∆ analogously. Then EΓ and E∆ do not have Boreldisjoint complete sections.

Proof. Let A be any Borel subset of Free(NΓ∗∆). Then the range of thef produced by Lemma 2.1 is either an EΓ-invariant set contained in A, oran E∆-invariant set contained in the complement of A. Hence, A cannotsimultaneously meet every E∆ class and have its complement meet everyEΓ-class. �

We now use this fact to obtain a couple of results on matchings and edgecolorings of Borel bipartite graphs.

Theorem 3.8. For every n > 1, there exists an n-regular acyclic Borelbipartite graph with no Borel perfect matching.

A DETERMINACY APPROACH TO BOREL COMBINATORICS 13

Proof. Let Γ = ∆ = Z/nZ in Theorem 3.7. Let Y ⊆ [Free(NΓ∗∆)]n be thestandard Borel space consisting of the equivalence classes of EΓ and E∆.Let G be the intersection graph on Y . This is an n-regular acyclic Borelbipartite graph. If M ⊆ Y × Y was a Borel perfect matching for G, thensetting

A = {x ∈ NΓ∗∆ : ∃(R,S) ∈M such that {x} = R ∩ S},we see that A and the complement of A would be Borel disjoint completesections for EΓ and E∆, contradicting Theorem 3.7. �

The graph used above was suggested as a candidate for a graph with noperfect matching by Conley and Kechris [3]. Lyons and Nazarov [20] haveshown that in the case n = 3, this graph has a measurable matching withrespect to a natural measure.

Theorem 3.9. For every n, there exists an n-regular acyclic Borel bipartitegraph with no Borel edge coloring with 2n− 2 colors.

Proof. We use the same graph as in Theorem 3.8. Suppose for a contradic-tion that it had a Borel edge coloring with 2n− 2 colors. By the pigeonholeprinciple, each vertex of G must be incident to at least one edge assigned aneven color, and at least one edge assigned an odd color. Let A be the set ofpoints x in Free(NΓ∗∆) such that {x} = R∩S where R is an equivalence classof EΓ, S is an equivalence class of E∆, and the edge (R,S) in G is coloredwith an even color. Then A is a complete section for EΓ, and the complementof A is a complete section for E∆, contradicting Theorem 3.7. �

Now we can give an exact characterization of the possible Borel edgechromatic numbers of n-regular acyclic Borel bipartite graphs.

Theorem 3.10. For every n ≥ 1 and every m ∈ {n, . . . , 2n − 1}, there isan n-regular acyclic Borel bipartite graph G such that χ′B(G) = m.

Proof. Let G be an n-regular acyclic Borel graph on X with χ′(G) ≥ 2n− 1by Theorem 3.9. Let m be an element of {n, . . . , 2n− 1}. There is an edgecoloring of G using 2n− 1 colors by Theorem 1.1 and our discussion in theintroduction before Theorem 1.4. Let G′ ⊆ G be the set of edges coloredusing one of the first m colors. Then clearly the graph G′ on X has a Boreledge coloring with m colors. It cannot have a Borel edge coloring with m−1colors as this would give an edge coloring of G with 2n− 2 colors. Now letY be an uncountable standard Borel space, and let H be an extension of G′

to an n-regular Borel bipartite graph H on X tY such that each connectedcomponent of H \G′ has at most one point in X. Then χ′B(H) = m. �

In the theorems we have proved above, we have mostly worked on spacesof the form Free(NΓ). As we described in the introduction, this is quitenatural since the graph G(Γ,N) achieves the maximal chromatic number ofall Borel graphs generated by free actions of Γ. However, it is interesting toask what happens when we change our base space to be finite. For example,

14 ANDREW S. MARKS

it is an open question whether there is a dichotomy characterizing when apair of countable Borel equivalence relations admits Borel disjoint completesections, and here we would like to know whether Theorem 3.7 remains truewhen we change N to be some finite k. As we will see, this is the case whenk = 3, but it is open for k = 2. Likewise, we would like to compute theBorel chromatic number of graphs of the form G(Γ, k) for k ≥ 2. Clearly, ifk ≤ m are both at least 2, then χB(G(Γ, k)) ≤ χB(G(Γ,m)) ≤ χB(G(Γ,N)).It is open whether these chromatic numbers can ever be different1 :

Question 3.11. Does there exist a finitely generated marked group Γ suchthat χB(G(Γ,N)) 6= χB(G(Γ, 2))?

Certainly, there are no obvious tools to show such chromatic numberscan be different. One approach to showing that these chromatic numbersare always the same would be to show the existence of a Borel homomor-phism from G(Γ,N) to G(Γ, 2). To do this it would be sufficient to find anequivariant Borel function from Free(NΓ) to Free(2Γ). We note that such afunction could not be injective in the case when Γ is sofic (which includesall the examples of groups we have discussed). This follows from results ofBowen on sofic entropy, as pointed out by Thomas [25, Theorem 6.11].

In the measurable context, when (X,µ) is a standard probability space, wecan say a bit more about the µΓ-measurable chromatic number of graphs ofthe form G(Γ, X), as X and µ vary. Recall from [3] that the µ-measurablechromatic number of a Borel graph G on a standard probability space(X,µ) is the least cardinality of a Polish space Y such that there is a µ-measurable coloring c : X → Y of G. Now given Borel actions a and b of Γon the Borel probability spaces (X,µ) and (Y, ν) respectively, a factor mapfrom a to b is a µ-measurable equivariant function f : X → Y such that thepushforward of µ under f is ν. Bowen [1, Theorem 1.1] has shown that if Γcontains a nonabelian free subgroup, then given any nontrivial probabilitymeasures µ and ν on the standard Borel spaces X and Y , there is a factormap from the left shift action of Γ on (XΓ, µΓ) to the left shift action ofΓ on (Y Γ, νΓ). Hence, the µΓ-measurable chromatic number of G(Γ, X) isequal to the νΓ-measurable chromatic number of G(Γ, Y ) for all such (X,µ)and (Y, ν). For some more results of this type for nonamenable groups ingeneral, see [2].

We now return to the pure Borel context, and end this section by notingthat we have the following variant of Lemma 2.1 for finite base spaces.This lemma can be proved using a nearly identical argument to that ofLemma 2.1 except changing the application of the pigeon-hole principle inthe obvious way. From this, one can derive versions of all of the Theoremsabove for finite base spaces. For example, we have χB(G(Γ∗∆,m+n−1)) ≥

1Recently, Seward and Tucker-Drob [24] have answered this question in the negative.They show that for every countable group Γ, there is an equivariant Borel function fromFree(NΓ) → Free(2Γ).

A DETERMINACY APPROACH TO BOREL COMBINATORICS 15

χB(G(Γ,m)) + χB(G(∆, n)) − 1, and Theorem 3.7 and its corollaries holdusing 3 instead of N.

Lemma 3.12. Let Γ,∆ be countable groups and m,n ≥ 2 be finite. IfA ⊆ Free((m+n−1)Γ∗∆) is any Borel set, then at least one of the followingholds:

(1) There is an continuous injective function f : Free(nΓ)→ Free((m+n− 1)Γ∗∆) that is equivariant with respect to the left shift action ofΓ on these spaces and such that ran(f) ⊆ A.

(2) There is an continuous injective function f : Free(m∆)→ Free((m+n− 1)Γ∗∆) that is equivariant with respect to the left shift action of∆ on these spaces and such that ran(f) ⊆ Ac.

4. Disjoint complete sections for measure and category

We turn now to the question of whether Theorem 3.7 can be proved usingpurely measure theory or Baire category. In the case when Γ = ∆ = Z/2Z,we can prove Theorem 3.7 using either of these two tools. If the generatorsof Γ and ∆ are α and β, then any nontrivial product probability measureon Free(N(Z/2Z)∗(Z/2Z)) has the property that the map x 7→ αβ ·x is ergodic,and the two maps x 7→ α · x and x 7→ β · x are both measure preserving.This is enough to conclude the Theorem 3.7 in this case. We can similarlygive a Baire category argument using generic ergodicity. We will show thatΓ = ∆ = Z/2Z is the only nontrivial pair of Γ and ∆ for which measure orcategory can prove Theorem 3.7.

We begin by showing that Borel disjoint complete sections exist in themeasure context for aperiodic countable Borel equivalence relations. Recallthat an equivalence relation is said to be aperiodic if all of its equivalenceclasses are infinite.

Lemma 4.1. Let µ be a Borel probability measure on a standard Borel spaceX. Then if E and F are aperiodic countable Borel equivalence relationson X, there exist disjoint Borel sets A and B such that A meets µ-a.e.equivalence class of E and B meets µ-a.e. equivalence class of F .

Proof. It follows from the marker lemma [18, Lemma 6.7] that we can finda decreasing sequence C0 ⊇ C1 ⊇ . . . of Borel sets that are each completesections for both E and F and such that their intersection

⋂Ci is empty.

Note that for each n and ε > 0, there is i > n such that

µ([Cn \ Ci]E) > 1− ε and µ([Cn \ Ci]F ) > 1− ε

It follows that we can find a strictly increasing sequence (ik)k such that

µ([⋃k≥0

(Ci2k \ Ci2k+1)]E) = 1 and µ([

⋃k≥0

(Ci2k+1\ Ci2k+2

)]F ) = 1

Now set A = ∪k≥0(Ci2k \ Ci2k+1) and B = ∪k≥0(Ci2k+1

\ Ci2k+2). �

16 ANDREW S. MARKS

Our goal is to extend this result to all pairs of countable Borel equivalencerelations E and F where every E class has at least 2 elements and everyF -class has at least 3 elements. We will do this by reducing it to the casewe have already proved above. More precisely, in Theorem 4.5 we will showthat several types of problems are equivalent in a Borel way to the problemof finding Borel disjoint complete sections for pairs of such equivalence rela-tions. That is, to each instance of each type of problem, we will demonstratehow to construct an instance of each of the other types so that a solutionto these problems can be transformed in a Borel way into a solution of theoriginal problem. The exact sense in which this is done will be clear in ourproof. Of course, the idea of reductions between combinatorial problems hasa long history. For an example of recent work with a similar effective flavor,see [10].

We first introduce another combinatorial problem. If G is a graph on X,an antimatching of G is a function f : X → X such that for all x ∈ X, wehave x G f(x) and f(f(x)) 6= x. A partial antimatching of G is a partialfunction f : X → X satisfying these conditions for all x ∈ dom(f).

We have the following lemma constructing antimatchings in the topolog-ical context, using a result of Conley and Miller on the existence of Borelmatchings in the topological context:

Lemma 4.2. Suppose n ≥ 3 and G is an acyclic Borel bipartite n-regulargraph on a Polish space X. Then there exists a Borel antimatching of Gmodulo a G-invariant meager set.

Proof. By [5], there exists a Borel perfect matching for G restricted to aG-invariant meager set C. Let A be one half of a Borel partition of Xwitnessing the bipartiteness of G, and let M be the Borel perfect matchingof G � C. Then we can construct a Borel antimatching f for G � C in thefollowing way: if x ∈ A and {x, y} ∈ M , then set f(x) = y. If x /∈ A, thenchoose some neighbor y of x such that {x, y} /∈M and set f(x) = y. �

The following lemma is useful when dealing with Borel antimatchings.

Lemma 4.3. Suppose G is a locally countable Borel graph, and f is a partialBorel antimatching of G such that ran(f) ⊆ dom(f), and every connectedcomponent of G contains some x ∈ dom(f). Then f can be extended to atotal Borel antimatching f∗ of G.

Proof. Define f∗ as follows. Let f∗(x) = f(x) if x ∈ dom(f). Otherwise,let f∗(x) = y, for some neighbor y of x such that the distance in G fromy to an element of dom(f) is as small as possible (using Lusin-Novikovuniformization [17, 18.10, 18.15] to choose such a y when there is more thanone). Then clearly f∗(f∗(x)) 6= x since for any x /∈ dom(f), we have thatf∗(x) is closer to some element of dom(f) than x. �

Throughout this section, we assume that we have a Borel linear order onall our standard Borel spaces. Thus, when we speak of the least element of

A DETERMINACY APPROACH TO BOREL COMBINATORICS 17

some finite subset of a standard Borel space, we are referring to the leastelement with respect to this order. One way of obtaining such a linearorder is via a Borel bijection with a standard Borel space equipped with acanonical Borel linear ordering, such as the one on R. These linear orderingsare useful when we need to break “ties” in our constructions when we arefaced with some irrelevant choice. In cases where we need to choose oneof finitely many points, we will generally break ties by choosing the leastpoint according to this ordering. In cases where we need to choose one ofcountably many options, we can use uniformization as we have above.

Lemma 4.4. If G is an acyclic locally finite Borel graph of degree ≥ 2,then there is a partial Borel antimatching f of G such that G � (dom(f))c

is 2-regular.

Proof. Let G be a locally finite Borel graph of degree ≥ 2 on a standardBorel space X. Using Lemma 2.2, let {Ai}i∈N be a Borel partition of Xsuch that for all i, for all distinct x, y ∈ Ai, the distance between x and y inG is greater than 2.

Let k0, k1, . . . be a sequence containing each natural number infinitelymany times. We define a sequence f0 ⊆ f1 ⊆ . . . of partial Borel antimatch-ings whose union will be the f we desire. These fi will all have the propertythat if x ∈ ran(fi) and x /∈ dom(fi), then there exist exactly two neighborsy of x such that y /∈ dom(fi) or fi(y) 6= x.

Let f0 = ∅. Now we define fi+1 ⊇ fi. For each x ∈ Aki such thatx /∈ dom(fi), do the following: if there exists some neighbor y of x such thaty ∈ dom(fi) and fi(y) 6= x, then use uniformization to choose some such yand define fi+1(x) = y. If there does not exist any such y and x /∈ ran(fi),then choose exactly two neighbors y1 and y2 of x and define fi+1(y) = x forall neighbors y of x that are not equal to y1 or y2.

Let f =⋃i∈N fi. Now if x /∈ dom(f), there are exactly two neighbors y

of x such that y /∈ dom(f) or f(y) 6= x. However, if x had a neighbor y suchthat f(y) 6= x, then we would have x ∈ dom(f). Hence, both these two ymust not be in dom(f). Thus, G � (dom(f))c is 2-regular. �

We are now ready to proceed.

Theorem 4.5. Suppose n ≥ 3. Then the following statements are all false.However, the statements are all true modulo a nullset with respect to anyBorel probability measure, and true modulo a meager set with respect to anycompatible Polish topology.

(1) Every pair E and F of countable Borel equivalence relations on astandard Borel space X such that the E-classes all have cardinality≥ 3 and the F -classes all have cardinality ≥ 2 admits disjoint Borelcomplete sections.

(2) Every pair E and F of independent aperiodic countable Borel equiv-alence relations admits disjoint Borel complete sections.

18 ANDREW S. MARKS

(3) Every locally finite Borel graph G having degree at least 3 has a Borelantimatching.

(4) Every acyclic Borel bipartite n-regular graph G has a Borel anti-matching.

Proof. (1) is false by Theorem 3.7. (2) is true in the measure-theoreticcontext by Lemma 4.1. (4) is true in the topological context by Lemma 4.2.

We will finish the proof of the theorem by showing (1) ⇒ (2) ⇒ (3) ⇒(1), and (3) ⇒ (4) ⇒ (2). Further, each of these implications will be donein a “local” way so that these implications also yield the truth of thesestatements in the measure and category contexts. We will discuss this morein what follows.

(1) ⇒ (2) is obvious.(2) ⇒ (3). Let X be a standard Borel space. We will begin by proving

the special case where G is a 3-regular acyclic Borel graph on {0, 1} × Xwhere (0, x) G (1, y) if and only if x = y. For i ∈ {0, 1}, let Fi be theequivalence relation on X such that x Fi y if and only if (i, x) and (i, y) arein the same connected component of G � {i} ×X. The Fi are independentbecause G is acyclic. Let B ⊆ X be a Borel set such that B is a completesection for F0 and Bc is a complete section for F1. We can use B to define aBorel antimatching. The rough idea is to direct elements of {0}×X towardselements of B and direct elements of {1} ×X away from elements of B.

If x ∈ B, define f((0, x)) = (1, x). Then let z be a point of Bc suchthat (1, z) is closest to (1, x) in G � {1} × X (breaking ties as usual), anddefine f((1, x)) = (1, y) where (1, y) is the neighbor of (1, x) along the pathfrom (1, x) to (1, z). Likewise, if x ∈ Bc, define f((1, x)) = (0, x), let z bea point of B such that (0, z) is closest to (0, x) in G � {0} ×X, and definef((0, x)) = (0, y) where (0, y) is the neighbor of (0, x) along the path from(0, x) to (0, z).

Now let G be an arbitrary locally finite Borel graph on X having degree atleast 3. First, we may assume that G is acyclic. To see this, use Lemma 2.2to obtain a Borel set C of pairwise disjoint cycles that contains at least onecycle from each connected component of G containing a cycle. Now definea Borel antimatching f on these connected components as follows. For eachcycle x0, x1, . . . xn = x0 in C, let f(xi) = xi+1 for i < n, and f(xn) = x0.Now use Lemma 4.3 to extend f to a total Borel antimatching f∗ on theseconnected components.

So assume that G is acyclic. By Lemma 4.4, we can find a partial Borelantimatching of G such that G � (dom(f))c is 2-regular. Let A = (dom(f))c.Now take a Borel set of edges of G � A that are pairwise disjoint and so thatthe set contains at least one edge from each connected component of G � A.Remove these edges from G to obtain the Borel graph G′ on X. Now usingLemma 4.4 on G′, we may obtain another set B ⊆ X that is the complementof a partial Borel antimatching on G′ such that G′ � B is 2-regular. Note

A DETERMINACY APPROACH TO BOREL COMBINATORICS 19

that G � A and G′ � B do not have any connected components that areequal.

Now these A and B correspond to places where we have failed to constructantimatchings. Hence, without loss of generality, we may assume that eachconnected component of G meets both A and B. By Lemma 2.2, let C be aBorel set of pairwise disjoint finite paths in G from elements of A to elementsof B that contains at least one path from every connected component of G.We may assume that if x0, . . . , xn is a path in C, then x0 is the only pointof this path in A, and xn is the only point of this path in B. (We allowpaths consisting of a single point where A and B intersect). Thus, each pairof connected components of G � A and G′ � B are connected by at most onepath in C, since G is acyclic.

Let S ⊆ X consist of the connected components of G � A that meetonly finitely many paths in C. Since this set has a Borel transversal, wecan obtain a Borel antimatching of G � S. We can then use Lemma 4.3to extend this to a Borel antimatching of the connected components of Gthat meet S. An identical comment is true for B. Thus, without loss ofgenerality, we can assume that for each connected component of G � A andG′ � B, if there is a path in C that meets this connected component, thenthere are infinitely many.

Let Y be the collection of starting points of paths in C, and Z be the col-lection of ending points of paths in C, so there is a canonical Borel bijectionbetween Y and Z. Note that Y and Z may have nonempty intersection.Define W = {0} × Y ∪ {1} × Z. Consider the 3-regular Borel graph H onW , defined by the following three conditions. First, (0, x) H (1, y) if andonly if there is a path in C from x to y. Second, (0, x) H (0, y) if and only ifthere is a path from x to y in G � A that does not contain any other elementof Y . Third, (1, x) H (1, y) if and only if there is a path from x to y inG′ � B that does not contain any other element of Z. H is 3-regular sinceconnected components of G � A and G′ � B that are met by paths in C aremet by infinitely many such paths.H is a graph of the type we discussed at the beginning of this proof, and

hence we can find a Borel antimatching of H. Let A∗ ⊆ A be the pointsthat are in the same connected component of G � A as some element ofY . Let B∗ ⊆ B be the points that are in the same connected componentof G′ � B as some element of Z. It is clear that we can lift the Borelantimatching of H to a partial Borel antimatching f of G whose domain isA∗ ∪ B∗ ∪ {x : ∃p ∈ C(x ∈ p)}, and such that ran(f) ⊆ dom(f). We finishby applying Lemma 4.3.

Our proof above has shown that (2) ⇒ (3). We now show that assumingthat (2) is true modulo a nullset with respect to every Borel probabilitymeasure implies that (3) is true modulo a nullset with respect to everyBorel probability measure.

Assume G is a locally finite Borel graph on X and µ is a Borel probabilitymeasure on X. Let EG be the connectedness relation for G. We can find

20 ANDREW S. MARKS

a Borel probability measure ν which dominates µ and such that ν is EG-quasi-invariant [18, Section 8]. Now perform the same process as above toobtain a pair of equivalence relations E and F on some Borel subset Y of X,such that from Borel disjoint complete sections for E and F , we can definea Borel antimatching of G.

Now this transformation of disjoint complete sections for E and F intoan antimatching of G is “local” in the sense that inside each connectedcomponent C of G, we have a Borel way of transforming disjoint completesections for E � Y ∩C and F � Y ∩C into an antimatching of G � C. Hence,given disjoint Borel sets A and B such that A meets ν-a.e. E-class and Bmeets ν-.a.e. F -class, we can find a Borel antimatching of G restricted to aBorel ν-conull set, since ν is EG-quasi-invariant.

Throughout the remainder of this proof, the same idea as above can beused to turn pure Borel implications between our four statements into im-plications in the measure context, and in the Baire category context. Weleave it to the reader to perform the rest of these transformations.

(3)⇒ (1) Let E and F be countable Borel equivalence relations such thatevery E-class has cardinality ≥ 3 and every F class has cardinality ≥ 2.By [18, Proposition 7.4], there exist Borel equivalence relations E∗ ⊆ E andF ∗ ⊆ F such that every E∗-class is finite and has cardinality ≥ 3 and everyF ∗-class is finite and has cardinality ≥ 2. Hence, we may assume that allthe equivalence classes of E and F are finite. Let Y tZ be the disjoint unionof the equivalence classes of E and the equivalence classes of F respectively.Let G be the graph on Y t Z where R and S are adjacent in G if R ∈ Y ,S ∈ Z, and R ∩ S 6= ∅.

Now let W be an uncountable standard Borel space, and extend G to alocally finite Borel graph G∗ on Y t Z tW so that every vertex in Y tWhas degree ≥ 3 in G∗, every vertex in Z has degree ≥ 2 in G∗, and such thatR ∈ Y t Z is adjacent to an element of W in G∗ if and only if R ∈ Y andthe degree of R is < 3 in G or R ∈ Z and the degree of R is < 2 in G. Notethat for such R there must be S ∈ Y t Z distinct from R such that R ∩ Shas cardinality ≥ 2.

Now let f be a Borel antimatching of G∗. Of course, G∗ does not havedegree ≥ 3. However, the neighbors of every degree 2 vertex in G∗ all havedegree ≥ 3. Hence, we can contract away vertices of degree 2, find a Borelantimatching of this graph, and then use it in the obvious way to find aBorel antimatching of G∗.

Let A0 be the set of x ∈ X such that there exists R ∈ Y such thatf(R) ∈ Z and R ∩ f(R) = {x}. Let A1 be the Borel set of x ∈ X such thatthere exists an R ∈ Y and S ∈ Z such that R ∩ S has cardinality ≥ 2, andx is the least element of R ∩ S. Let A = A0 ∪ A1. Clearly A meets everyequivalence class of E, and Ac meets every equivalence class of F .

(3) ⇒ (4) is obvious.

A DETERMINACY APPROACH TO BOREL COMBINATORICS 21

(4) ⇒ (2). Suppose we have two independent aperiodic countable Borelequivalence relations E and F on a standard Borel space X. By [18, Propo-sition 7.4] we can find E∗ and F ∗, finite Borel subequivalence relations ofE and F whose equivalence classes all have cardinality n. The intersectiongraph of their equivalence classes is Borel bipartite and n-regular. From aBorel antimatching for this graph, we can produce Borel disjoint completesections for E and F , as in the proof that (3) ⇒ (1). �

Our final goal will be to prove a theorem about edge colorings for 3-regularacyclic Borel bipartite graphs in the context of measure and category. Thiswill follow from several more equivalences extending those of Theorem 4.5above.

We begin with another definition. Suppose G is a graph on X. A direct-ing of G is a set D ⊆ G that contains exactly one of (x, y) and (y, x) forevery pair of neighbors x, y ∈ X. A partial directing of G is a subset ofG that contains at most one of (x, y) and (y, x) for every pair of neighborsx, y ∈ X. Given a partial directing D of a graph G, say that a point x ∈ Xis a source if (x, y) ∈ D for some y, and (y, x) /∈ D for all y. Similarly, saythat a point x ∈ X is a sink if (y, x) ∈ D for some y, and (x, y) /∈ D for ally. Of course, if f is an antimatching of a graph G and we extend the set{(x, f(x)) : x ∈ X} to a directing D of G, then this directing will have nosinks.

Lemma 4.6. Suppose that G is a locally countable Borel graph such thateach vertex of G has degree ≥ 2, and D is a partial Borel directing of Gwithout sources or sinks. Suppose also that every connected component of Gcontains at least one vertex that is incident to an edge of D. Then D canbe extended to a total Borel directing D∗ of G that has no sources or sinks.

Proof. Suppose that x0, x1, . . . , xn is a path in G such that x0 and xn areboth incident to edges already in D. Then we can extend D by adding theedges from x0, x1, . . . , xn that do not conflict with edges already in D; add(xi, xi+1) to D unless (xi+1, xi) is already in D. The property that D hasno sources or sinks is preserved when we add paths in this way. Similarly,given a cycle, we can extend D using this cycle in the analogous way, whilepreserving the property that D has no sources or sinks.

Use Lemma 2.2 to partition all the finite paths and cycles of G intocountably many Borel sets {Pi}i∈N such that the elements of each Pi arepairwise disjoint. Let k0, k1, . . . be a sequence that contains each elementof N infinitely many times. Let D0 = D. Now define Di+1 from Di byextending Di via all the cycles of Pki , and all the paths of Pki that start andend at vertices incident to at least one edge in Di. Let D∞ =

⋃i∈NDi.

Let A be the set of vertices that are incident to at least one edge in D∞.It is clear that if x0 ∈ A and x0, x1, . . . , xn is a path in G, then xn ∈ Aimplies that xi ∈ A for all 0 ≤ i ≤ n.

We finish by extending D∞ to D∗ by directing the remaining edges ofG “away” from D∞. More precisely, let x and y be distinct elements of X

22 ANDREW S. MARKS

and suppose that neither (x, y) nor (y, x) are in D∞. Then there must be aunique path x0, . . . , xn such that x0 is incident to an edge in D∞, and thepath ends with (xn−1, xn) equal to (y, x) or (x, y). Extend D∞ to D∗ byadding all such (xn−1, xn). �

Theorem 4.7. The fallowing statements are all false in the full Borel con-text. They are true modulo a nullset with respect to any Borel probabilitymeasure, and true modulo a meager set with respect to any compatible Polishtopology.

(1) Every pair of countable Borel equivalence relations E and F on astandard Borel space X such that the E-classes all have cardinality≥ 3 and the F -classes all have cardinality ≥ 2 admits disjoint Borelcomplete sections.

(2) For every pair of aperiodic countable Borel equivalence relations Eand F on a standard Borel space X, there exists a Borel set B ⊆ Xsuch that B and Bc are complete sections for both E and F .

(3) Every 3-regular Borel graph has a directing with no sinks or sources.(4) Every 3-regular Borel bipartite graph has a Borel edge coloring with

4 colors.

Proof. (1) is false by Theorem 3.7 and true in the measure and categorycontext by Theorem 4.5. We will use the same type of proof as Theorem 4.5.

(1) ⇒ (2). Since E and F are aperiodic, the argument that (3) ⇒ (1) inTheorem 4.5 produces subequivalence relations E∗ and F ∗ of E and F withfinite classes, and a Borel set A such that A and Ac are complete sections forE∗, and Ac is a complete section for F ∗. Hence, A meets every E-class, andAc meets every E-class and every F -class in infinitely many places. Thus,if we run the same argument on the aperiodic equivalence relations E � Ac

and F � Ac with their roles reversed, we obtain a Borel set A′ ⊆ Ac suchthat A′ meets every F � Ac-class, and (A′)c meets both every E � Ac-classand every F � Ac-class. Now let B = A ∪A′.

(2) ⇒ (1) follows from Theorem 4.5.(2) ⇒ (3). Let G be a 3-regular Borel graph. Using Lemma 4.6, we may

assume that G is acyclic, as in the proof of (2) ⇒ (3) for Theorem 4.5.We begin by letting Y ⊆ [X]2 be a Borel set of pairwise disjoint edges of

G that contains at least one edge from each connected component of G. Wedefine two countable Borel equivalence relations E and F on Y as follows:R and S are related by E if their least points are connected in G \ Y , andrelated by F if their greatest points are connected in G \ Y . Here we useG \ Y to denote the graph G with the edges from Y removed.

We may assume that all the equivalence classes of E and F are infinite;on the connected components of G\Y that correspond to equivalence classesof E and F that are finite, and we can apply Lemma 4.6 to get a directingof the connected components of G containing points corresponding to finiteE-classes or F -classes.

A DETERMINACY APPROACH TO BOREL COMBINATORICS 23

Now let B ⊆ Y be a Borel set such that both B and Bc are completesections for E and F . Let D0 = {(x, y) : {x, y} ∈ B and x is less than y}.Each connected component of G \ Y contains infinitely many x such that(x, y) ∈ D0 for some y, and infinitely many y such that (x, y) ∈ D0 forsome x. We will extend D0 to a total Borel directing of G without sinks orsources.

Consider the set of paths x0, x1, . . . , xn in G \ Y such that there exists yand z such that both (y, x0) and (xn, z) are in D0. We may use Lemma 2.2 topartition these paths into countably many Borel sets {Pi}i∈N such that theelements of each Pi are pairwise disjoint. Now as in the proof of Lemma 4.6,for each i ∈ N, extend each Di to Di+1 by adding the edges from the pathsof Pi which do not conflict with edges already in Di. Let D∞ =

⋃i∈NDi.

Then complete D∞ to a total directing D using Lemma 4.6.(3) ⇒ (1) follows from Theorem 4.5. It is clear that such a directing can

be used to define a Borel antimatching.(3) ⇒ (4). Suppose that G is a Borel bipartite 3-regular graph whose

bipartiteness is witnessed by the Borel sets A and B. Suppose D is a Boreldirecting of G without sinks or sources. We can use D to write G as thedisjoint union of two graphs H0 and H1 in the following way: the edges ofH0 are those directed by D from A to B, and the edges of H1 are thosedirected by D from B to A. The vertices in H0 and H1 all have degree 1or 2. Hence, each connected component of the Hi is finite, a ray (havingexactly one vertex of degree 1), or a line (having no vertices of degree 1).

If all the connected components of H0 and H1 were finite or rays, thenit would be trivial to construct a Borel edge coloring of G with four colors;we could simply edge color H0 using the colors {0, 1}, edge color H1 usingthe colors {2, 3}, and then take the union of these colorings. Our problemis that in general, we will need to use 3 colors in an edge coloring of an Hi

containing lines.Let Y ⊆ [X]2 be a Borel set of pairwise disjoint edges from H0 consist-

ing of infinitely many edges from each line in H0. Define the countableBorel equivalence relations F0 and F1 on Y where S and R are Fi-related ifthere exist x ∈ S and y ∈ R that are in the same connected component ofHi. Clearly every equivalence class of F0 is infinite, however, there may beequivalence classes of F1 that are finite.

Now take a Borel set C ⊆ Y that is a complete section for F0, so thatCc meets every infinite equivalence class of F1. We can find such a C byletting Z be an uncountable standard Borel space and extending F0 and F1

to aperiodic equivalence relations F ∗0 and F ∗1 on Y t Z such that if x ∈ Zand y ∈ Y , then x��F

∗0 y and xF ∗1 y only if [y]F1 is finite. Now find disjoint

complete sections for F ∗0 and F ∗1 .Let H∗0 be the graph H0 but with the edges from C removed, and let H∗1

be the graph H1 but with the edges from C added. Clearly H∗0 has no lines.Further, all the lines that we have added to H∗1 must contain rays from H1.This is because the elements of C in a new line in H∗1 must all be F1-related,

24 ANDREW S. MARKS

and therefore come from an F1-class that is finite. Hence, we can edge-colorthese lines from H∗1 in a Borel way with 2-colors.

If we perform the same process again with H∗1 and H∗0 in lieu of H0

and H1, respectively, then we obtain Borel graphs H∗∗0 and H∗∗1 such thatG = H∗∗0 ∪ H∗∗1 and both H∗∗0 and H∗∗1 have Borel edge colorings with 2colors.

(4) ⇒ (1). We use Theorem 4.5 again. Let G be a Borel bipartite 3-regular graph, whose bipartiteness is witnessed by the Borel sets A and B.Suppose that G has a Borel edge coloring with 4 colors. We can use thiscoloring to define a Borel antimatching of G. First, partition the four colorsinto the sets {0, 1} and {2, 3}. Notice that each vertex must be incident toat least one edge of color 0 or 1, and at least one edge of color 2 or 3. Thus,we can define a Borel antimatching by setting f(x) = y if x ∈ A and y isthe least neighbor of x such that (x, y) is colored 0 or 1, or if x ∈ B and yis the least neighbor of x such that (x, y) is colored 2 or 3. �

As a consequence of the above lemma, we obtain the following:

Theorem 4.8. Suppose G is a Borel bipartite 3-regular graph on X. ThenG has a Borel edge coloring with 4 colors modulo a null set or meager set withrespect to any Borel probability measure on X or Polish topology realizingthe standard Borel structure of X.

The full measurable analogue of Vizing’s theorem for Borel graphs remainsopen.

Question 4.9. Given any n-regular Borel graph G on a standard Borelprobability space (X,µ), must there be a µ-measurable edge coloring of Gwith n+ 1 colors?

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Department of Mathematics, California Institute of TechnologyE-mail address: marks@caltech.edu